Related papers: Subharmonicity of higher dimensional exponential t…

We study the phenomenon of cavitation for the displacement boundary value problem of radial, isotropic compressible elasticity for a class of stored energy functions of the form $W(F) + h(\det F)$, where $W$ grows like $||F||^n$, and $n$ is…

This paper is concerned with the Cauchy problem of the multi-dimensional incompressible magnetohydrodynamic equations with inhomogeneous density and fractional dissipation. It is shown that when $\alpha+\beta=1+\frac{n}{2}$ satisfying…

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, \alpha}$ for…

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic…

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…

It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the following property. For every $n\in \mathbb{N}$ and…

A basic building block in Classical Potential Theory is the fundamental solution of the Laplace equation in ${\mathbb R}^d$ (Newtonian kernel). The main goal of this article is to study the rates of nonlinear $n$-term approximation of…

We study the optimization of Steklov eigenvalues with respect to a boundary density function $\rho$ on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. We investigate the minimization and maximization of $\lambda_k(\rho)$, the…

We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite…

For all $n$, we define the $n$-dimensional critical catenoid $M_n$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in $\Bbb{R}^{n+1}$. We show that the…

Let $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(\Omega)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear…

We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law conservation law. When the…

We expand the classical balayage of measures and subharmonic functions on a system of rays $S$ with a common origin on the complex plane $\mathbb C$. This allows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$ build…

This paper models the theory of abstract harmonic spaces in the syntax of the continuous first-order logic of Banach lattices. It addresses a topological question asking when a one-to-one harmonic map onto smooth manifolds $M^n$ is a…

A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates u(x) =o(x_{n}^{1-alpha}|x|^{m+alpha})at infinity in the upper half space of Rn, which generalizes the growth properties of analytic…

This work provides a geometric characterization of the measures $\mu$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $R\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}\,d\mu(y)$ belongs…

Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a M\"obius invariant function bounding the norm of the Cauchy transform $\bf{C}$ from below. This function is…

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find…

Let $(K_{n})_{n=1}^{\infty}$ be the optimal constants satisfying the multilinear (real or complex) Bohnenblust--Hille inequality. The exact values of the constants $K_{n}$ are still waiting to be discovered since eighty years ago; recently,…

Symmetry problems in harmonic analysis are formulated and solved. One of these problems is equivalent to the refined Schiffer's conjecture which was recently proved by the author. Let $k=const>0$ be fixed, $S^2$ be the unit sphere in…